## Abstract

We demonstrate that the structure of complex second-order strongly elliptic operators H on R^{d} with coefficients invariant under translation by Z^{d} can be analyzed through decomposition in terms of versions H_{z}, z ∈ T^{d}, of H with z-periodic boundary conditions acting on L_{2}(I^{d}) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups S^{z} generated by the H_{z} have kernels with similar properties and z → S^{z} extends to a function on C^{d}\{0} which is analytic with respect to the trace norm. The sequence of semigroups S^{(m),z} obtained by rescaling the coefficients of H_{z} by c(x) → c(mx) converges in trace norm to the semigroup Ŝ^{z} generated by the homogenization Ĥ_{z} of H_{z}. These convergence properties allow asymptotic analysis of the spectrum of H.

Original language | English |
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Pages (from-to) | 621-650 |

Number of pages | 30 |

Journal | Mathematische Zeitschrift |

Volume | 232 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1999 |